The centrifugal force in a rotating frame can be elegantly expressed in tensor form as $F_c^i=T^{i j}{ }_x{ }^j$, where the tensor $T^{i j}=m\left(\omega^i \omega^j-\omega^2 \delta^{i j}\right)$ captures the dependence on angular velocity and mass. This formulation reveals that the force is symmetric and linear in position, acting radially outward from the axis of rotation. It vanishes precisely when the particle lies along the rotation axis, where $\vec{\omega} \times \vec{x}=0$, highlighting the geometric nature of fictitious forces in non-inertial frames.
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$\complement\cdots$Counselor
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Physical Interpretation:
The centrifugal force arises due to the rotation and acts radially outward from the axis of rotation. It depends on both the magnitude and direction of $\vec{x}$ relative to $\vec{\omega}$.
Force Vanishing Condition:
The centrifugal force vanishes when the particle lies along the axis of rotation, i.e., when $\vec{x} \| \vec{\omega}$. In this case, $\vec{\omega} \times \vec{x}=0$.
Geometric Insight:
The force is strongest when $\vec{x}$ is perpendicular to $\vec{\omega}$, and zero when aligned-highlighting the role of rotational geometry in fictitious forces.
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